ANOVA post hoc analysis¶

Post hoc analyses are currently in development.

The simple approaches described below will approximate appropriate post hoc results.

As a rule-of-thumb, NEVER report post hoc results which disagree with the main test results. If the results disagree, the post hoc procedures are too simple.

If you report post hoc results for spm1d analyses please explicitly highlight the limitations described in this document.

Warning

Current post hoc procedures in spm1d are likely too simple.

The example belowe uses a simple Bonferroni correction, but this assumes that the post hoc tests are independent, which is usually not the case.

Danger

Current post hoc procedures in spm1d are generally not valid.

They are not valid because they involve separate smoothness assessments for each post hoc test.

Although the resulting errors are expected to be small, we cannot guarantee validity.

Users are encouraged to explore other post hoc correction methods.

One-way ANOVA¶

This examples revisits the one-way example.

Since the one-way ANOVA results reached significance, we may justifiably conduct post hoc analyses.

For one-way ANOVA, post hoc analyses consist of two-sample t tests conducted on all group pairs.

Since there are three groups in our one-way ANOVA example, there are three pairs to test (1v2, 1v3, and 2v3).

A simple way to correct for multiple tests across these three group pairs is to use the Bonferroni correction:

```>>> alpha = 0.05
>>> nTests = 3
>>> p_critical = spm1d.util.p_critical_bonf(alpha, nTests)
```

The resulting critical p value is 0.016952.

Post hoc tests will be deemed significant if they reach the critical t value implied by the p = 0.016952.

Test statistics may be computed for the three group pairs as follows:

Note

The equal_var option should be the same as was selected for ANOVA.

```>>> t12   = spm1d.stats.ttest2(Y1, Y2, equal_var=False)
>>> t13   = spm1d.stats.ttest2(Y1, Y3, equal_var=False)
>>> t23   = spm1d.stats.ttest2(Y2, Y3, equal_var=False)
```

After computing the test statistics we can conduct inference using the corrected critical p value:

Warning

The two_tailed option must always be True in post hoc analyses; one-tailed inference is inconsistent with the null-hypothesis implied by ANOVA.

```>>> t12i  = t12.inference(alpha=p_critical, two_tailed=True)
>>> t13i  = t13.inference(alpha=p_critical, two_tailed=True)
>>> t23i  = t23.inference(alpha=p_critical, two_tailed=True)
```

In this case all tests reach significance.

These results may be interpreted exactly as in the case of a single t test: if there were truly no group differences, then smooth, random 1D continua would produce SPM{t}s that reaches the critical t threshold in fewer than alpha = 5% of many repeated experiments.

Warning

p values for post hoc tests.

If desired, you may report exact p values for each suprathreshold cluster, but in this case the p values must be adjusted for multiple comparisons.

To compute appropriate p values, use the inverse to the Bonferroni correction through spm1d.util.p_corrected_bonf, as described below.

Let’s consider the following result from one of the post hoc tests conducted above:

```SPM{T} inference field
SPM.z      :  (1x101) raw test stat field
SPM.df     :  (1, 33.641)
SPM.fwhm   :  4.82031
SPM.resels :  (1, 20.74555)
Inference:
SPM.alpha  :  0.017
SPM.zstar  :  4.05216
SPM.p      :  (<0.001, <0.001, <0.001, 0.008, <0.001)
```

Note

When multiple p values exist, they are listed in the following order: (1) upper threshold left-to-right, (2) lower threshold left-to-right. In the “t23i” case above (rightmost panel of the figure) there are five p values. The first corresponds to the single upper-threshold cluster. The next four correspond to the four lower-threshold clusters.

The five p values listed in the results above correspond to this test’s five supra-threshold clusters.

Corrected p values may be computed as follows:

```>>> p_value = 0.008
>>> nTests  = 3
>>> p_value_corrected = spm1d.util.p_corrected_bonf(p_value, nTests)
```

The corrected p value is 0.028. If this cluster had precisely touched the threshold, its p value would have been alpha.